p460 - 3d s.e.1 3d schrodinger equation simply substitute momentum operator do particle in box and h...
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P460 - 3D S.E. 1
3D Schrodinger Equation • Simply substitute momentum operator
• do particle in box and H atom
• added dimensions give more quantum numbers. Can have degeneracies (more than 1 state with same energy). Added complexity.
• Solve by separating variables
ti
m
x
tzyxVtzyx
orip
),,,(),,,(22
2
2
2
2
EzyxV
tzyxtzyx
m
),,(
)(),,(),,,(2
2
2
P460 - 3D S.E. 2
• If V well-behaved can separate further: V(r) or
Vx(x)+Vy(y)+Vz(z). Looking at second one:
• LHS depends on x,y RHS depends on z
• S = separation constant. Repeat for x and y
zz
yxyx
yxyx
zyx
zz
zyx
yxyx
zyx
zyxm
VEVV
VEVV
zyxzyxassume
EzVyVxV
z
z
2
2
2
2
2
2
2
2
2
2
2
2
2
)()(
)()()(
)()()(),,(
))()()((22
SVV
SVE
yxyxyx
zdz
d
yx
z
z
)(
)(
2
2
2
2
2
2
1
P460 - 3D S.E. 3
• Example: 2D (~same as 3D) particle in a Square Box
• solve 2 differential equations and get
• symmetry as square. “broken” if rectangle
ESESSSEEE
ESEV
ESSV
ESV
zyx
zzdz
d
yydy
d
xxdx
d
z
z
y
y
x
x
)()'('
'
'
2
2
2
2
2
2
)()(),(
)()(
0
,0,,0
yxyx
yVxVVsatisfies
boxinsideV
ayyaxxV
yx
yx
)( 22
2 2
22
yxmayx nnEEE
P460 - 3D S.E. 4
• 2D gives 2 quantum numbers.
Level nx ny Energy
1-1 1 1 2E0
1-2 1 2 5E0
2-1 2 1 5E0
2-2 2 2 8E0
• for degenerate levels, wave functions can mix (unless “something” breaks degeneracy: external or internal B/E field, deformation….)
• this still satisfies S.E. with E=5E0
ionnormalizatdxdy
nnAyx
nnEEE
yxa
yn
axn
yxmayx
yx
1||
..2,1,sinsin),(
)(
2
22
2 2
22
1
sinsin
sinsin
222112
221
212
mix
ay
ax
ay
ax
A
A
P460 - 3D S.E. 5
Spherical Coordinates
• Can solve S.E. if V(r) function only of radial coordinate
• volume element is
• solve by separation of variables
• multiply each side by
ErVr
r
rErV
r
rrrrM
M
)(),,(]
)(sin)([
),,()(
2
2
22
22
2
2
sin1
sin12
2
22
)sin)(()( drrddrvold
R
R
rRr
r
rrr
rr
MRVE
2
2
22
22
2
22
sin1
sin
sin12)( )(
)()()(),,(
Rr 22 sin
P460 - 3D S.E. 6
Spherical Coordinates-Phi
• Look at phi equation first
• constant (knowing answer allows form)
• must be single valued
• the theta equation will add a constraint on the m quantum number
21 ),()(2
2
ldd mrf
lime )(
.......2,1,0
)()2()2(
limim mee ll
P460 - 3D S.E. 7
Spherical Coordinates-Theta
• Take phi equation, plug into (theta,r) and rearrange
• knowing answer gives form of constant. Gives theta equation which depends on 2 quantum numbers.
• Associated Legendre equation. Can use either analytical (calculus) or algebraic (group theory) to solve. Do analytical. Start with Legendre equation
)1()(
)]([)(
sinsin1
sin
21
2
2
2
222
ll
rVE
dd
ddm
rMdrdRr
drd
R
l
)1()( 2
2
sin
sinsin
1 lllmdd
dd
functionLegendrePz
Pllzz
l
ldzdP
dz
Pd ll
cos
0)1(2)1( 2
22
P460 - 3D S.E. 8
Spherical Coordinates-Theta
• Get associated Legendre functions by taking the derivative of the Legendre function. Prove by substitution into Legendre equation
• Note that power of P determines how many derivatives one can do.
• Solve Legendre equation by series solution
2
2
0 1
1
2
)1(
0)1(2)1(
2
2
2
2
k
kkdz
Pd
k k
kkdz
dPkkl
ldzdP
dz
Pd
zkka
kzazaP
Pllzz ll
dzPd
dzdP
dz
Pdmlm
z
z
lml
lml
l
z
z
P
z
2
21
||
||
)1(
)1(
)1(
222
122
21
220
2/||2
P460 - 3D S.E. 9
Solving Legendre Equation
• Plug series terms into Legendre equation
• let k-1=j+2 in first part and k=j in second (think of it as having two independent sums). Combine all terms with same power
• gives recursion relationship
• series ends if a value equals 0 L=j=integer
• end up with odd/even (Parity) series
0})]1()1([)1({ 2 kk
kk zallkkzakk
0})]1()1([)1)(2{( 2j
jj zalljjajj
jjjlljj
j aa )1)(2()1()1(
2
)1()1(02 lljja j
0,00,0 01 oddeven aaoraa
P460 - 3D S.E. 10
Solving Legendre Equation
• Can start making Legendre polynomials. Be in ascending power order
• can now form associated Legendre polynomials. Can only have l derivatives of each Legendre polynomial. Gives constraint on m (theta solution constrains phi solution)
2212
60
)1)(2()1()1(
210
110
010
313
,0,1,2
1,0,1
10,1,0
zP
aaal
zPaal
Paal
jjlljj
lm
Pz
l
ldzdm
lm lm
lml
l
||
)1( ||
||2/||2
P460 - 3D S.E. 11
Spherical Harmonics
• The product of the theta and phi terms are called Spherical Harmonics. Also occur in E&M.
• They hold whenever V is function of only r. Seen related to angular momentum
)1(
)1(
31
)1(
1
22,2
21,2
220
21,1
10
00
21
21
z
zz
z
z
z
harmonicsspherical
Y mlmlm
P460 - 3D S.E. 12
3D Schr. Eqn.-Radial Eqn.
• For V function of radius only. Look at radial equation
• can be rewritten as (usually much better...)
• note L(L+1) term. Angular momentum. Acts like repulsive potential and goes to infinity at r=0 (ala classical mechanics)
• energy eigenvalues typically depend on 2 quantum numbers (n and L). Only 1/r potentials depend only on n (and true for hydrogen atom only in first order. After adding perturbations due to spin and relativity, depends on n and j=L+s).
Rr
llRErV
dr
dRr
dr
d
r 22
2 )1())(
21
)()(
))1(
2(
2 2
2
2
22
rrRru
Euur
llV
dr
ud
dru
drrRrP2
22
4
4)(
P460 - 3D S.E. 13
Particle in spherical box
• Good first model for nuclei
• plug into radial equation. Can guess solutions
• look first at l=0
• boundary conditions. R=u/r and must be finite at r=0. Gives B=0. For continuity, must have R=u=0 at r=a. gives sin(ka)=0 and
• note plane wave solution. Supplement 8-B discusses scattering, phase shifts. General terms are
)cos()sin(
222
2
krBkrAu
kwithuk ME
drud
rarn
an
Man
n nE)/sin(
21
00
20 ....3,2,12
222
r
erR
rki
)(
arrV
arrV
)(
0)(
)()())1(
2(
2 2
2
2
22
rrRruEuur
llV
dr
ud
P460 - 3D S.E. 14
Particle in spherical box
• ForLl>0 solutions are Bessel functions. Often arises in scattering off spherically symmetric potentials (like nuclei…..). Can guess shape (also can guess finite well)
• energy will depend on both quantum numbers
• and so 1s 1p 1d 2s 2p 2d 3s 3d …………….and ordering (except higher E for higher n,l) depending on details
• gives what nuclei (what Z or N) have filled (sub)shells being different than what atoms have filled electronic shells. In atoms:
• in nuclei (with j subshells)
.....222120121110 EEEEEEEnl
PSS
NeBeHeZ
221
)(1042
21
25
21
23 21111
)(1614862
sdpps
SSiOCHeZ
P460 - 3D S.E. 15
H Atom Radial Function
• For V =a/r get (use reduced mass)
• Laguerre equation. Solutions are Laguerre polynomials. Solve using series solution (after pulling out an exponential factor), get recursion relation, get eigenvalues by having the series end……n is any integer > 0 and L<n. Energy doesn’t depend on L quantum number.
• Where fine structure constant alpha = 1/137 used. Same as Bohr model energy
• eigenfunctions depend on both n,L quantum numbers. First few:
2
2
2
222
2220
42 6.1322)4( n
eVZn
Zcm
neMZ
neE
0
0
0
0
2
200
2/21
2/20
40
/10
)2(
5.0
aZraZr
aZraZr
em
aZr
eR
eR
AaeR C
e
Rr
llRE
r
Zem
dr
dRr
dr
d
r 20
2
2
2 )1(
4
21
P460 - 3D S.E. 16
H Atom Wave Functions
P460 - 3D S.E. 17
H Atom Degeneracy
• As energy only depends on n, more than one state with same energy for n>1
• ignore spin for now
Energy n l m D
-13.6 eV 1 0(S) 0 1
-3.4 eV 2 0 0 1
1(P) -1,0,1 3
-1.5 eV 3 0 0 1
1 -1,0,1 3
2(D) -2,-1,0,1,2 5
1 Ground State
4 First excited states
9 second excited states
2nD
P460 - 3D S.E. 18
Probability Density
• P is radial probability density
• small r naturally suppressed by phase space (no volume)
• can get average, most probable radius, and width (in r) from P(r). (Supplement 8-A)
22
2
0
221
10
2
0
22
00
2
2
||)(
cos||
sin||
1||
||
nlRrrP
drddror
drddr
ionnormalizatdVolume
yprobabilit
22
0
rrrwidthrraverage
probablemost drdP
P460 - 3D S.E. 19
Most probable radius• For 1S state
• Bohr radius (scaled for different levels) is a good approximation of the average or most probable value---depends on n and L
• but electron probability “spread out” with width about the same size
0204
920
20
/2222
)1(21
023
0
0
/22/2
/2222
87.03
3
))]1(1[(
)(
)"("
20
||)(
0
20
2
0
0
20
0
aaar
adrerArr
generalin
adrrrPr
peakar
ere
eArRArrP
ar
n
llZan
ararar
drdP
ar
P460 - 3D S.E. 20
Radial Probability Density
P460 - 3D S.E. 21
Radial Probability Density
note # nodes
P460 - 3D S.E. 22
Angular Probabilities
• no phi dependence. If (arbitrarily) have phi be angle around z-axis, this means no x,y dependence to wave function. We’ll see in angular momentum quantization
• L=0 states are spherically symmetric. For L>0, individual states are “squished” but in arbitrary direction (unless broken by an external field)
• Add up probabilities for all m subshells for a given L get a spherically symmetric probability distribution
1||
)sin(|)(||)(|),(2
22
im
m e
P
"1"
sincos
"1"
211
21,1
210
21110
00
statesPA
statesSA
P460 - 3D S.E. 23
Orthogonality
• each individual eigenfunction is also orthogonal.
• Many relationships between spherical harmonics. Important in, e.g., matrix element calculations. Or use raising and lowering operators
• example
mlmnmmllnn
mlnnlm
Rwith
dddrrmlnnlm
'''
2'''
0 0
2
0
* sin
1''0)(
'''|cos|
cos
cos||
ˆtan
)1'('
10
llmmrf
mlnrnlm
polynomialLegendreis
noterEV
zintconsE
llmm
P460 - 3D S.E. 24
Wave functions• build up wavefunctions from eigenfunctions.
• example
• what are the expectation values for the energy and the total and z-components of the angular momentum?
• have wavefunction in eigenfunction components
)2(6
1),,,( /
121/
211/
100221 tiEtiEtiE eeetr
dvolt
idvolHHE **|
6
1)120(
6
1)2(
6
1
1))11(1)11(12)10(0(6
1
))1()1(2)1((6
124
7)
4
3(
6
1)2(
6
1
110
1111002
111221
zzzz LLLL
llllllL
EEEEEEE